Tensorial
4.0 and
TContinuumMechanics 2.1
The package TContinuumMechanics
is devoted to
the manipulation of tensors in the context of Continuum Mechanics
problems. The most recent version is TContinuumMechanics
2.1, which needs Mathematica
(version 4 or 5) and Tensorial 4.0,
has been divided into two parts: the program TContinuumMechanics
version 2.1
itself, which is now a shareware as is version 4.0 of Tensorial,
and its specific
applications which remain freely available.
The various
functions created for the present purpose are shown in the
link Package.
The main example
of application TContinuumMechanics_Fluegge is composed
of the twelve chapters of the well known book of Wilhelm
Flügge "Tensorial Analysis and Continuum Mechanics", Wilhelm
Flügge, Springer, 1972. It shows how TContinuumMechanics
2.1 can be used
in continuum mechanics problems. In this application, I closely
followed Flügge's book all along the chapters, as it is to my
opinion one of the clearest presentation of Continuum Mechanics using
tensors extensively. It demonstrates how the Mathematica
packages Tensorial
and TContinuumMechanics
permit to
express in a very clear manner the tensorial structure of the equations
of Continuum Mechanics.

New
in Tensorial
4.0 and
ContinuumMechanics
2.0 - January 2006
All the output can now be copied and
pasted. See
UsingTensorial, Output Format.
The output notation has been improved. Unexpanded partial derivatives
now show only a single comma before the first differentiation index.
Similarly an unexpanded covariant derivative shows only a single
semicolon. (Other differentiation symbols can be used as before.) In
addition there is an alternative mode of display for covariant
derivatives using the \[Del] symbol with subscripts. This can be turned
on and off with SetCovariantDisplay. Similarly, SetLieDisplay can be
used to display unexpanded Lie derivatives as a £ symbol with
subscripts.
The old Dif and Cov wrappers for
differentiation
indices are out. Unexpanded (to coordinates) partial and covariant
derivatives are only broken out by their linear and Liebnizian
properties and then left without further evaluation. This means that
the old NestedTensor is unnecesssary and it has been removed. To
prevent linear and Liebnizian breakout wrap the expression in Tensor
instead of NestedTensor. UnnestTensor will still unwrap the expression.
DummySimplify has been removed. There is
a new
TensorSimplify. It selects terms that have the same pattern, applies
any declared symmetries and then applies SimplifyTensorSum on these
subsets of terms.
The Curvature section contains new
routines (moved
from the General Relativity subpackage) that calculate the Riemann
tensor, up and down, the Ricci tensor, the scalar curvature and the
Einstein tensor.
The new OrthonormalTransformation routine will, given a metric and a
signature pattern, calculate a transformation matrix from the
coordinated basis to an orthonormal basis.
A number of routines that are more in the class of general expression
manipulation rather than tensor routines are gathered together in the
Functions & Rules section. Two new routines here are
SymbolsToPatterns and LHSSymbolsToPatterns. These are useful in turning
derived equations into general rules that can be applied to subsequent
expressions.
The $PrePrint for larger font output has
been
eliminated. Instead the Tensorial style sheet was changed to give a
larger Output cell font. This simplifies copying and pasting by
eliminating additional box structures. The style sheet was also changed
to eliminate the Helvetica font in Headings in favor of Times. Some
systems have a difficult time using the Helvetica font.
Also, in the Tensorial style sheet,
Output cells are
now StandardForm. Derivatives are formated in Leibnizian style but you
will now have to use MatrixForm to format
arrays.
Many improvements have been made to
various
functions of the program. I have added as an example a chapter showing
how TContinuumMechanics can be used to calculate classical
operators like divergence, gradient, curl or laplacian in arbitrary
curvilinear coordinates. Practical examples will be progressively added
in the future, even if the present notebook shows that
TContinuumMechanics (associated to Tensorial ) already allows
to
cover all the various aspects of continuum mechanics.
Later on I plan to develop the
modelization of
growing hulls and membranes with tensorial surface tensions, following
the course of Jean Garrigues (Ecole Supérieure de
Mécanique de Marseille). Some preliminary works have already
started in our group, involving liquid crystal and fibered type of
structures.
The development of this package led to
the creation
of operations and rules which are included in part in
`TContinuumMechanics`Help`3D package`, in part in Tensorial. It shows
how convenient is the symbolic approach which permits now to manipulate
the whole basic concepts presented for instance in the standard
Flügge's book.
Figures presented in Flügge's
book to clarify
the theoretical developments, have been introduced here using the
Mathematica package `DrawGraphics`. This program may be
downloaded on the same web site as TContinuumMechanics and Tensorial.New
in
ContinuumMechanics
2.1 - May 2008
There was an error in the flavor
parameter of the functions RaiseIndexD, LowerIndexD and
UpDownSwapD, which has been corrected.

Acknowledgments
I would like to thank Renan Cabrera, the
author of
Tensorial, and his collaborator David Park, to have
accepted my contribution to the improvement of the successive versions
of Tensorial, and to have themselves participated to the amendment of
the present package.